1 Introduction
Machine learning is very different from human learning. Humans are able to learn from very few labeled examples and apply the learned knowledge to new examples in novel conditions. In contrast, traditional machine learning algorithms assume that the training and test data are independent and identically distributed (i.i.d.) and supervised machine learning methods only perform well when the given extensive labeled data are from the same distribution as the test distribution. However, this assumption rarely holds in practice, as the data are likely to change over time and space. To compensate for the degradation in performance due to domain shift, domain adaptation saenko2010adapting ; sa ; gfk ; long_cvpr ; bmvc15 ; coral ; tzeng_arxiv15 ; daume tries to transfer knowledge from source (training) domain to target (test) domain. Our approach is in line with most existing unsupervised domain adaptation approaches in that we first transform the source data to be as close to the target data as possible. Then a classifier trained on the transformed source domain is applied to the target data.
In this chapter, we mainly focus on the unsupervised scenario as we believe that leveraging the unlabeled target data is key to the success of domain adaptation. In real world applications, unlabeled target data are often much more abundant and easier to obtain. On the other side, labeled examples are very limited and require human annotation. So the question of how to utilize the unlabeled target data is more important for practical visual domain adaptation. For example, it would be more convenient and applicable to have a pedestrian detector automatically adapt to the changing visual appearance of pedestrians rather than having a human annotator label every frame that has a different visual appearance.
Our goal is to propose a simple yet effective approach for domain adaptation that researchers with little knowledge of domain adaptation or machine learning could easily integrate into their application. In this chapter, we describe a “frustratingly easy” (to borrow a phrase from daume ) unsupervised domain adaptation method called CORrelation ALignment or CORAL coral in short. CORAL aligns the input feature distributions of the source and target domains by minimizing the difference between their secondorder statistics. The intuition is that we want to capture the structure of the domain using feature correlations. As an example, imagine a target domain of Western movies where most people wear hats; then the “head” feature may be positively correlated with the “hat” feature. Our goal is to transfer such correlations to the source domain by transforming the source features.
In Section 2
, we describe a linear solution to CORAL, where the distributions are aligned by recoloring whitened source features with the covariance of the target data distribution. This solution is simple yet efficient, as the only computations it needs are (1) computing covariance statistics in each domain and (2) applying the whitening and recoloring linear transformation to the source features. Then, supervised learning proceeds as usual–training a classifier on the transformed source features.
For linear classifiers, we can equivalently apply the CORAL transformation to the classifier weights, leading to better efficiency when the number of classifiers is small but the number and dimensionality of the target examples are very high. We present the resulting CORAL–Linear Discriminant Analysis (CORALLDA) corallda in Section 3, and show that it outperforms standard Linear Discriminant Analysis (LDA) by a large margin on cross domain applications. We also extend CORAL to work seamlessly with deep neural networks by designing a layer that consists of a differentiable CORAL loss dcoral , detailed in Section 4. On the contrary to the linear CORAL, Deep CORAL learns a nonlinear transformation and also provides endtoend adaptation. Section 5 describes extensive quantitative experiments on several benchmarks, and Section 6 concludes the chapter.
2 Linear Correlation Alignment
In this section, we present CORrelation ALignment (CORAL) for unsupervised domain adaptation and derive a linear solution. We first describe the formulation and derivation, followed by the main linear CORAL algorithm and its relationship to existing approaches. In this section and Section 3, we constrain the transformation to be linear. Section 4 extends CORAL to learn a nonlinear transformation that aligns correlations of layer activations in deep neural networks (Deep CORAL).
2.1 Formulation and Derivation
We describe our method by taking a multiclass classification problem as the running example. Suppose we are given sourcedomain training examples , with labels , , and target data , . Here both and are the dimensional feature representations of input . Suppose and
are the feature vector means and covariance matrices. Assuming that all features are normalized to have zero mean and unit variance,
after the normalization step, while .To minimize the distance between the secondorder statistics (covariance) of the source and target features, we apply a linear transformation to the original source features and use the Frobenius norm as the matrix distance metric:
(1)  
where is covariance of the transformed source features and denotes the squared matrix Frobenius norm.
If , then an analytical solution can be obtained by choosing such that . However, the data typically lie on a lower dimensional manifold outlooks ; gfk ; sa , and so the covariance matrices are likely to be low rank who . We derive a solution for this general case, using the following lemma.
Lemma 1
Let be a real matrix of rank and X a real matrix of rank at most , where ; let be the SVD of , and , , be the largest singular values and the corresponding left and right singular vectors of respectively. Then, is the optimal solution to the problem of . SVT
Theorem 2.1
Let be the MoorePenrose pseudoinverse of , and denote the rank of and respectively. Then, is the optimal solution to the problem in Equation (12) with .
Proof
Since is a linear transformation, does not increase the rank of . Thus, . Since and are symmetric matrices, conducting SVD on and gives and respectively. We first find the optimal value of through considering the following two cases:
Case 1
. The optimal solution is . Thus, is the optimal solution to Equation (12) where .
Combining the results in the above two cases yields that is the optimal solution to Equation (12) with . We then proceed to solve for based on the above result. Letting , we can write
Since , we have
This gives:
Let , then the right hand side of the above equation can be rewritten as . This gives
By setting to , we obtain the optimal solution of as
(2)  
2.2 Algorithm
Figures 1(ac) illustrate the linear CORAL approach. Figure 1(a) shows example original source and target data distributions. We can think of the transformation intuitively as follows: the first part whitens the source data, while the second part recolors it with the target covariance. This is illustrated in Figure 1(b) and Figure 1(c) respectively.
In practice, for the sake of efficiency and stability, we can avoid the expensive SVD steps and perform traditional data whitening and coloring. Traditional whitening adds a small regularization parameter to the diagonal elements of the covariance matrix to explicitly make it full rank and then multiplies the original features by its inverse square root (or square root for coloring.) This is advantageous because: (1) it is faster^{1}^{1}1the entire CORAL transformation takes less than one minute on a regular laptop for dimensions as large as and and more stable, as SVD on the original covariance matrices might not be stable and might be slow to converge; (2) as illustrated in Figure 2, the performance is similar to the analytical solution in Equation (2) and very stable with respect to . In the experiments provided at the end of this chapter we set to 1. The final algorithm can be written in four lines of MATLAB code as illustrated in Algorithm 1.
One might instead attempt to align the distributions by whitening both source and target. As shown in Figure 1(d), this will fail as the source and target data are likely to lie on different subspaces due to domain shift. An alternative approach would be whitening the target and then recoloring it with the source covariance. However, as demonstrated in outlooks ; sa and our experiments, transforming data from source to target space gives better performance. This might be due to the fact that by transforming the source to target space the classifier is trained using both the label information from the source and the unlabelled structure from the target.
After CORAL transforms the source features to the target space, a classifier parametrized by can be trained on the adjusted source features and directly applied to target features. For a linear classifier , we can apply an equivalent transformation to the parameter vector (e.g., ) instead of the features (e.g., ). This results in added efficiency when the number of classifiers is small but the number and dimensionality of target examples is very high. For linear SVM, this extension is straightforward. In Section 3, we apply the idea of CORAL to another commonly used linear classifier–Linear Discriminant Analysis (LDA). LDA is special in the sense that its weights also use the covariance of the data. It is also extremely efficient for training a large number of classifiers who .
2.3 Relationship to Existing Methods
It has long been known that input feature normalization improves many machine learning methods, e.g., batchnorm
. However, CORAL does not simply perform feature normalization, but rather aligns two different distributions. Batch Normalization
batchnorm tries to compensate for internal covariate shift by normalizing each minibatch to be zeromean and unitvariance. However, as illustrated in Figure 1(a), such normalization might not be enough. Even if used with full whitening, Batch Normalization may not compensate for external covariate shift: the layer activations will be decorrelated for a source point but not for a target point. What’s more, as mentioned in Section 2.2, whitening both domains is not a successful strategy.Recent stateoftheart unsupervised approaches project the source and target distributions into a lowerdimensional manifold and find a transformation that brings the subspaces closer together gopalaniccv11 ; gfk ; sa ; outlooks . CORAL avoids subspace projection, which can be costly and requires selecting the hyperparameter that controls the dimensionality of the subspace, . We note that subspacemapping approaches outlooks ; sa only align the top eigenvectors of the source and target covariance matrices. On the contrary, CORAL aligns the covariance matrices, which can only be reconstructed using all eigenvectors and eigenvalues. Even though the eigenvectors can be aligned well, the distributions can still differ a lot due to the difference of eigenvalues between the corresponding eigenvectors of the source and target data. CORAL is a more general and much simpler method than the above two as it takes into account both eigenvectors and eigenvalues of the covariance matrix without the burden of subspace dimensionality selection.
Maximum Mean Discrepancy (MMD) based methods (e.g., TCA tca , DAN dan_long15
) for domain adaptation can be interpreted as “moment matching” and can express arbitrary statistics of the data. Minimizing MMD with a polynomial kernel (
with ) is similar to the CORAL objective, however, no previous work has used this kernel for domain adaptation nor proposed a closed form solution to the best of our knowledge. The other difference is that MMD based approaches usually apply the same transformation to both the source and target domain. As demonstrated in ref:kulis_cvpr11 ; outlooks ; sa , asymmetric transformations are more flexible and often yield better performance for domain adaptation tasks. Intuitively, symmetric transformations find a space that “ignores” the differences between the source and target domain while asymmetric transformations try to “bridge” the two domains.3 CORAL Linear Discriminant Analysis
In this section, we introduce how CORAL can be applied for aligning multiple linear classifiers. In particular, we take LDA as the example for illustration, considering LDA is a commonly used and effective linear classifier. Combining CORAL and LDA gives a new efficient adpative learning approach CORALLDA. We use the task of object detection as a running example to explain CORALLDA.
We begin by describing the decorrelationbased approach to detection proposed in who . Given an image , it follows the slidingwindow paradigm, extracting a dimensional feature vector at each window across all locations and at multiple scales. It then scores the windows using a scoring function
(3) 
In practice, all windows with values of above a predetermined threshold are considered positive detections.
In recent years, use of the linear SVM as the scoring function , usually with Histogram of Gradients (HOG) as the features , has emerged as the predominant object detection paradigm. Yet, as observed by Hariharan et al. who , training SVMs can be expensive, especially because it usually involves costly rounds of hard negative mining. Furthermore, the training must be repeated for each object category, which makes it scale poorly with the number of categories.
Hariharan et al. who proposed a much more efficient alternative, learning with Linear Discriminant Analysis (LDA). LDA is a wellknown linear classifier that models the training set of examples with labels as being generated by .
is the prior on class labels and the classconditional densities are normal distributions
(4) 
where the feature vector covariance is assumed to be the same for both positive and negative (background) classes. In our case, the feature is represented by . The resulting classifier is given by
(5) 
The innovation in who was to reuse and , the background mean, for all categories, reducing the task of learning a new category model to computing the average positive feature, . This was accomplished by calculating and
for the largest possible window and subsampling to estimate all other smaller window sizes. Also,
was shown to have a sparse local structure, with correlation falling off sharply beyond a few nearby image locations.Like other classifiers, LDA learns to suppress nondiscriminative structures and enhance the contours of the object. However it does so by learning the global covariance statistics once for all natural images, and then using the inverse covariance matrix to remove the nondiscriminative correlations, and the negative mean to remove the average feature. LDA was shown in who to have competitive performance to SVM, and can be implemented both as an exemplarbased exemplarsvm or as deformable parts model (DPM) dpm .
We observe that estimating global statistics and once and reusing them for all tasks may work when training and testing in the same domain, but in our case, the source training data is likely to have different statistics from the target data. Figure 4 illustrates the effect of centering and decorrelating a positive mean using global statistics from the wrong domain. The effect is clear: important discriminative information is removed while irrelevant structures are not.
Based on this observation, we propose an adaptive decorrelation approach to detection. Assume that we are given labeled training data in the source domain (e.g., virtual images rendered from 3D models), and unlabeled examples in the target domain (e.g., real images collected in an office environment). Evaluating the scoring function in the source domain is equivalent to first decorrelating the training features , computing their positive and negative class means and and then projecting the decorrelated feature onto the decorrelated difference between means, , where . This is illustrated in Figure 3(ab).
However, as we saw in Figure 4, the assumption that the input is properly decorrelated does not hold if the input comes from a target domain with a different covariance structure. Figure 3(c) illustrates this case, showing that does not have isotropic covariance. Therefore, cannot be used directly.
We may be able to compute the covariance of the target domain on the unlabeled target points , but not the positive class mean. Therefore, we would like to reuse the decorrelated mean difference , but adapt to the covariance of the target domain. In the rest of the chapter, we make the assumption that the difference between positive and negative means is the same in the source and target. This may or may not hold in practice, and we discuss this further in Section 5.
Let the estimated target covariance be . We first decorrelate the target input feature with its inverse square root, and then apply directly, as shown in Figure 3(d). The resulting scoring function is:
(6)  
This corresponds to a transformation instead of the original whitening being applied to the difference between means to compute . Note that if source and target domains are the same, then equals to since both and are symmetric. In this case, Equation 6 ends up the same as Equation 5.
In practice, either the source or the target component of the above transformation may also work, or even statistics from similar domains. However, as we will see in Section 5.2, dissimilar domain statistics can significantly hurt performance. Furthermore, if either source or target has only images of the positive category available, and cannot be used to properly compute background statistics, the other domain can still be used.
CORALLDA works in a purely unsupervised way. Here, we extend it to semisupervised adaptation when a few labeled examples are available in the target domain. Following ICRA14 , a simple adaptation method is used whereby the template learned on source positives is combined with a template learned on target positives, using a weighted combination. The key difference with our approach is that the target template uses targetspecific statistics.
In ICRA14 , the author uses the same background statistics as who which were estimated on 10,000 natural images from the PASCAL VOC 2010 dataset. Based on our analysis above, even though these background statistics were estimated from a very large amount of real image data, it will not work for all domains. In section 5.2, our results confirm this claim.
4 Deep CORAL
In this section, we extend CORAL to work seamlessly with deep neural networks by designing a differentiable CORAL loss. Deep CORAL enables endtoend adaptation and also learns more a powerful nonlinear transformation. It can be easily integrated into different layers or network architectures. Figure 5 shows a sample Deep CORAL architecture using our proposed correlation alignment layer for deep domain adaptation. We refer to Deep CORAL as any deep network incorporating the CORAL loss for domain adaptation.
We first describe the CORAL loss between two domains for a single feature layer. Suppose the numbers of source and target data are and respectively. Here both and are the dimensional deep layer activations of input that we are trying to learn. Suppose () indicates the th dimension of the th source (target) data example and () denote the feature covariance matrices.
We define the CORAL loss as the distance between the secondorder statistics (covariances) of the source and target features:
(7) 
where denotes the squared matrix Frobenius norm. The covariance matrices of the source and target data are given by:
(8) 
(9) 
where 1 is a column vector with all elements equal to 1.
The gradient with respect to the input features can be calculated using the chain rule:
(10) 
(11) 
In our experiments, we use batch covariances and the network parameters are shared between the two networks, but other settings are also possible.
To see how this loss can be used to adapt an existing neural network, let us return to the multiclass classification problem. Suppose we start with a network with a final classification layer, such as the ConvNet shown in Figure 5
. As mentioned before, the final deep features need to be both discriminative enough to train a strong classifier and invariant to the difference between source and target domains. Minimizing the classification loss itself is likely to lead to overfitting to the source domain, causing reduced performance on the target domain. On the other hand, minimizing the CORAL loss alone might lead to degenerated features. For example, the network could project all of the source and target data to a single point, making the CORAL loss trivially zero. However, no strong classifier can be constructed on these features. Joint training with both the classification loss and CORAL loss is likely to learn features that work well on the target domain:
(12) 
where denotes the number of CORAL loss layers in a deep network and is a weight that trades off the adaptation with classification accuracy on the source domain. As we show below, these two losses play counterparts and reach an equilibrium at the end of training, where the final features are discriminative and generalize well to the target domain.
5 Experiments
We evaluate CORAL and Deep CORAL on object recognition saenko2010adapting using standard benchmarks and protocols. In all experiments we assume the target domain is unlabeled. For CORAL, we follow the standard procedure sa ; decaf and use a linear SVM as the base classifier. The model selection approach of sa is used to set the parameter for the SVM by doing crossvalidation on the source domain. For CORALLDA, as efficiency is the main concern, we evaluate it on the more time constrained task–object detection. We follow the protocol of ICRA14 and use HOG features. To have a fair comparison, we use accuracies reported by other authors with exactly the same setting or conduct experiments using the source code provided by the authors.
5.1 Object Recognition
In this set of experiments, domain adaptation is used to improve the accuracy of an object classifier on novel image domains. Both the standard Office saenko2010adapting and extended OfficeCaltech10 gfk datasets are used as benchmarks in this chapter. OfficeCaltech10 contains 10 object categories from an office environment (e.g., keyboard, laptop, etc.) in 4 image domains: , , , and . The standard Office dataset contains 31 (the same 10 categories from OfficeCaltech10 plus 21 additional ones) object categories in 3 domains: , , and .
AC  AD  AW  CA  CD  CW  DA  DC  DW  WA  WC  WD  AVG  
NA  35.8  33.1  24.9  43.7  39.4  30.0  26.4  27.1  56.4  32.3  25.7  78.9  37.8 
SVMA  34.8  34.1  32.5  39.1  34.5  32.9  33.4  31.4  74.4  36.6  33.5  75.0  41.0 
DAM  34.9  34.3  32.5  39.2  34.7  33.1  33.5  31.5  74.7  34.7  31.2  68.3  40.2 
GFK  38.3  37.9  39.8  44.8  36.1  34.9  37.9  31.4  79.1  37.1  29.1  74.6  43.4 
TCA  40.0  39.1  40.1  46.7  41.4  36.2  39.6  34.0  80.4  40.2  33.7  77.5  45.7 
SA  39.9  38.8  39.6  46.1  39.4  38.9  42.0  35.0  82.3  39.3  31.8  77.9  45.9 
CORAL  40.3  38.3  38.7  47.2  40.7  39.2  38.1  34.2  85.9  37.8  34.6  84.9  46.7 
Object Recognition with Shallow Features
We follow the standard protocol of gfk ; sa ; gopalaniccv11 ; ref:kulis_cvpr11 ; saenko2010adapting and conduct experiments on the OfficeCaltech10 dataset with shallow features (SURF). The SURF features were encoded with 800bin bagofwords histograms and normalized to have zero mean and unit standard deviation in each dimension. Since there are four domains, there are 12 experiment settings, namely, AC (train classifier on (A)mazon, test on (C)altech), AD (train on (A)mazon, test on (D)SLR), AW, and so on. We follow the standard protocol and conduct experiments in 20 randomized trials for each domain shift and average the accuracy over the trials. In each trial, we use the standard setting gfk ; sa ; gopalaniccv11 ; ref:kulis_cvpr11 ; saenko2010adapting and randomly sample the same number (20 for , , and ; 8 for as there are only 8 images per category in the domain) of labelled images in the source domain as training set, and use all the unlabelled data in the target domain as the test set.
In Table 1, we compare our method to five recent published methods: SVMA svma , DAM ref:duan09 , GFK gfk , SA sa , and TCA tca
as well as the no adaptation baseline (NA). GFK, SA, and TCA are manifold based methods that project the source and target distributions into a lowerdimensional manifold. GFK integrates over an infinite number of subspaces along the subspace manifold using the kernel trick. SA aligns the source and target subspaces by computing a linear map that minimizes the Frobenius norm of their difference. TCA performs domain adaptation via a new parametric kernel using feature extraction methods by projecting data onto the learned transfer components. DAM introduces smoothness assumption to enforce the target classifier share similar decision values with the source classifiers. Even though these methods are far more complicated than ours and require tuning of hyperparameters (
e.g., subspace dimensionality), our method achieves the best average performance across all the 12 domain shifts. Our method also improves on the no adaptation baseline (NA), in some cases increasing accuracy significantly (from 56% to 86% for DW).Object Recognition with Deep Features
For visual domain adaptation with deep features, we follow the standard protocol of gfk ; dan_long15 ; decaf ; tzeng_arxiv15 ; reversegrad and use all the labeled source data and all the target data without labels on the standard Office dataset saenko2010adapting . Since there are 3 domains, we conduct experiments on all 6 shifts (5 runs per shift), taking one domain as the source and another as the target.
In this experiment, we apply the CORAL loss to the last classification layer as it is the most general case–most deep classifier architectures (e.g.
, convolutional neural networks, recurrent neural networks) contain a fully connected layer for classification. Applying the CORAL loss to other layers or other network architectures is also possible. The dimension of the last fully connected layer (
) was set to the number of categories (31) and initialized with . The learning rate ofwas set to 10 times the other layers as it was training from scratch. We initialized the other layers with the parameters pretrained on ImageNet
imagenet and kept the original layerwise parameter settings. In the training phase, we set the batch size to 128, base learning rate to , weight decay to , and momentum to 0.9. The weight of the CORAL loss () is set in such way that at the end of training the classification loss and CORAL loss are roughly the same. It seems be a reasonable choice as we want to have a feature representation that is both discriminative and also minimizes the distance between the source and target domains. We used Caffe
caffe and BVLC Reference CaffeNet for all of our experiments.We compare to 7 recently published methods: CNN alexnet (no adaptation), GFK gfk , SA sa , TCA tca , CORAL coral , DDC tzeng_arxiv15 , DAN dan_long15 . GFK, SA, and TCA are manifold based methods that project the source and target distributions into a lowerdimensional manifold and are not endtoend deep methods. DDC adds a domain confusion loss to AlexNet alexnet and finetunes it on both the source and target domain. DAN is similar to DDC but utilizes a multikernel selection method for better mean embedding matching and adapts in multiple layers. For direct comparison, DAN in this paper uses the hidden layer . For GFK, SA, TCA, and CORAL, we use the feature finetuned on the source domain ( in coral ) as it achieves better performance than generic pretrained features, and train a linear SVM sa ; coral . To have a fair comparison, we use accuracies reported by other authors with exactly the same setting or conduct experiments using the source code provided by the authors.
From Table 2 we can see that Deep CORAL (DCORAL) achieves better average performance than CORAL and the other 6 baseline methods. In 3 out of 6 shifts, it achieves the highest accuracy. For the other 3 shifts, the margin between DCORAL and the best baseline method is very small ().
AD  AW  DA  DW  WA  WD  AVG  
GFK  52.40.0  54.70.0  43.20.0  92.10.0  41.80.0  96.20.0  63.4 
SA  50.60.0  47.40.0  39.50.0  89.10.0  37.60.0  93.80.0  59.7 
TCA  46.80.0  45.50.0  36.40.0  81.10.0  39.50.0  92.20.0  56.9 
CORAL  65.70.0  64.30.0  48.50.0  96.10.0  48.20.0  99.80.0  70.4 
CNN  63.80.5  61.60.5  51.10.6  95.40.3  49.80.4  99.00.2  70.1 
DDC  64.40.3  61.80.4  52.10.8  95.00.5  52.20.4  98.50.4  70.6 
DAN  65.80.4  63.80.4  52.80.4  94.60.5  51.90.5  98.80.6  71.3 
DCORAL  66.80.6  66.40.4  52.80.2  95.70.3  51.50.3  99.20.1  72.1 
Domain Adaptation Equilibrium
To get a better understanding of Deep CORAL, we generate three plots for domain shift AW. In Figure 6(a) we show the training (source) and testing (target) accuracies for training with v.s. without CORAL loss. We can clearly see that adding the CORAL loss helps achieve much better performance on the target domain while maintaining strong classification accuracy on the source domain.
In Figure 6(b) we visualize both the classification loss and the CORAL loss for training w/ CORAL loss. As the last fully connected layer is randomly initialized with , in the beginning the CORAL loss is very small while the classification loss is very large. After training for a few hundred iterations, these two losses are about the same and reach an equilibrium. In Figure 6(c) we show the CORAL distance between the domains for training w/o CORAL loss (setting the weight to 0). We can see that the distance is getting much larger ( times larger compared to training w/ CORAL loss). Comparing Figure 6(b) and Figure 6(c), we can see that even though the CORAL loss is not always decreasing during training, if we set its weight to 0, the distance between source and target domains becomes much larger. This is reasonable as finetuning without domain adaptation is likely to overfit the features to the source domain. Our CORAL loss constrains the distance between source and target domain during the finetuning process and helps to maintain an equilibrium where the final features work well on the target domain.
5.2 Object Detection
Following protocol of ICRA14 , we conduct object detection experiment on the Office dataset saenko2010adapting with HOG features. We use the same setting as ICRA14 , performing detection on the Webcam domain as the target (test) domain, and evaluating on the same 783 image test set of 20 categories (out of 31). As source (training) domains, we use: the two remaining realimage domains in Office, Amazon and DSLR, and two domains that contain virtual images only, Virtual and VirtualGray, generated from 3d CAD models. The inclusion of the two virtual domains is to reduce human effort in annotation and facilitate future research sun16phdthesis . Examples of Virtual and VirtualGray are shown in Figure 8. Please refer to sun15virtualdataset for detailed explanation of the data generation process. We also compare to ICRA14 who use corresponding ImageNet imagenet synsets as the source. Thus, there are four potential source domains (two synthetic and three real) and one (real) target domain. The number of positive training images per category in each domain is shown in Table 3. Figure 7 shows an overview of our evaluation.
(a)  (b) 
Domain  # Training sample 

Amazon  20 
DSLR  8 
Virtual(Gray)  30 
ImageNet  1502000 
Effect of Mismatched Image Statistics
First, we explore the effect of mismatched precomputed image statistics on detection performance. For each source domain, we train CORALLDA detectors using the positive mean from the source, and pair it with the covariance and negative mean of other domains. The virtual and the Office domains are used as sources, and the test domain is always Webcam. The statistics for each of the four domains were calculated using all of the training data, following the same approach as who . The precomputed statistics of 10,000 real images from PASCAL, as proposed in who ; ICRA14 , are also evaluated.
Detection performance, measured in Mean Average Precision (MAP), is shown in Table 4. We also calculate the normalized Euclidean distance between pairs of domains as , and show the average distance between source and target in parentheses in Table 4.
From these results we can see a trend that larger domain difference leads to poorer performance. Note that larger difference to the target domain also leads to lower performance, confirming our hypothesis that both source and target statistics matter. Some of the variation could also stem from our assumption about the difference of means being the same not quite holding true. Finally, the PASCAL statistics from who perform the worst. Thus, in practice, statistics from either source domain or target domain or domains close to them could be used. However, unrelated statistics will not work even though they might be estimated from a very large amount of data as who .
Virtual  VirtualGray  Amazon  DSLR  PASCAL  

Virtual  30.8 (0.1)  16.5 (1.0)  24.1 (0.6)  28.3 (0.2)  10.7 (0.5) 
VirtualGray  32.3 (0.6)  32.3 (0.5)  27.3 (0.8)  32.7 (0.6)  17.9 (0.7) 
Amazon  39.9 (0.4)  30.0 (1.0)  39.2 (0.4)  37.9 (0.4)  18.6 (0.6) 
DSLR  68.2 (0.2)  62.1 (1.0)  68.1 (0.6)  66.5 (0.1)  37.7 (0.5) 
Unsupervised and Semisupervised Adaptation
Next, we report the results of our unsupervised and semisupervised adaptation technique. We use the same setting as ICRA14 , in which three positive and nine negative labeled images per category were used for semisupervised adaptation. Target covariance in Equation 6 is estimated from 305 unlabeled training examples. We also followed the same approach to learn a linear combination between the unsupervised and supervised model via crossvalidation. The results are presented in Table 5. Please note that our targetonly MAP is 52.9 compared to 36.6 in ICRA14 . This also confirms our conclusion that the statistics should come from a related domain. It is clear that both of our unsupervised and semisupervised adaptation techniques outperform the method in ICRA14 . Furthermore, VirtualGray data outperforms Virtual, and DSLR does best, as it is very close to the target domain (the main difference between and domains is in the camera used to capture images).
Finally, we compare our method trained on VirtualGray to the results of adapting from ImageNet reported by ICRA14 , in Figure 9. While their unsupervised models are learned from 1502000 real ImageNet images per category and the background statistics are estimated from 10,000 PASCAL images, we only have 30 virtual images per category and the background statistics is learned from about 1,000 images. What’s more, all the virtual images used are with uniform gray texturemap and white background. This clearly demonstrates the importance of domainspecific decorrelation, and shows that the there is no need to collect a large amount of real images to train a good classifier.
Source  Sourceonly who  UnsupOurs  SemiSup ICRA14  SemiSupOurs 
Virtual  10.7  27.9  30.7  45.2 
VirtualGray  17.9  33.0  35.0  54.7 
Amazon  18.6  38.9  35.8  53.0 
DSLR  37.7  67.1  42.9  71.4 
6 Conclusion
In this chapter, we described a simple, effective, and efficient method for unsupervised domain adaptation called CORrelation ALignment (CORAL). CORAL minimizes domain shift by aligning the secondorder statistics of source and target distributions, without requiring any target labels. We also developed novel domain adaptation algorithms by applying the idea of CORAL to three different scenarios. In the first scenario, we applied a linear transformation that minimizes the CORAL objective to source features prior to training the classifier. In the case of linear classifiers, we equivalently applied the linear CORAL transform to the classifier weights, significantly improving efficiency and classification accuracy over standard LDA on several domain adaptation benchmarks. We further extended CORAL to learn a nonlinear transformation that aligns correlations of layer activations in deep neural networks. The resulting Deep CORAL approach works seamlessly with deep networks and can be integrated into any arbitrary network architecture to enable endtoend unsupervised adaptation. One limitation of CORAL is that it captures secondorder statistics only and may not preserve higherorder structure in the data. However, as demonstrated in this chapter, it works fairly well in practice, and can also potentially be combined with other domainalignment loss functions.
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